Perturbative vacuum constraints in higher-curvature gravity: Schwarzschild deformations and strong-field observables
Higher-curvature terms modify the action but do not necessarily generate new vacuum geometries on the branch perturbatively connected to GR. We develop a first-order framework for static, spherical vacuum black holes in 4D metric theories with $\mathcal L_{\mathrm{grav}}=R/2+\lambda\Psi(R,X,Y)$, $X=R_{\mu\nu}R^{\mu\nu}$, and $Y=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$. On the coupling-analytic branch, purely Ricci-based terms $\Psi(R,X)$, analytic in curvature with $\Psi(0,0)=0$, do not deform a Ricci-flat background under adopted regularity and boundary assumptions. Riemann-dependent terms can, since Kretschmann scalar is nonzero. In areal-radius gauge, we derive model-independent first-order expressions for the horizon shift, ISCO, epicyclic frequencies, periapsis advance, photon sphere, critical shadow impact parameter, Wald entropy, and Hawking temperature. For $\Psi_\eta=\ell_\lambda^{-2+4\eta}\mathcal G^\eta$, treating noninteger powers as phenomenological parametrizations of nonlinear curvature, we obtain a closed first-order Schwarzschild-connected solution. Perturbations decay for $\eta>1/2$, while fixed-ADM interpretation requires $\eta>2/3$; at $\eta=2/3$, sourced $1/r$ term mixes with asymptotic mass mode. For positive effective coupling, the horizon, ISCO, photon sphere, and critical shadow impact parameter increase for $2/3<\eta<1$ and decrease for $\eta>1$. On the fixed-ADM branch, mass, Wald entropy, and Hawking temperature satisfy $dM=T_H,dS_{\rm W}$ to first order at fixed couplings. At $\eta=1$, the linear 4D Gauss--Bonnet term leaves local geometry and geodesic observables unchanged but adds a constant topological entropy shift. Finally, an Event Horizon Telescope-inspired shadow-size criterion gives a conservative first-order sensitivity estimate for the effective dimensionless coupling, a geometric consistency test rather than a complete observational constraint.